A polynomial-time approximation scheme for maximizing the minimum machine completion time

We discuss scheduling problems with m identical machines and n jobs where each job has to be assigned to some machine. The goal is to optimize objective functions that solely depend on the machine completion times. As a main result, we identify some conditions on the objective function, under which the resulting scheduling problems possess a polynomial-time approximation scheme. Our result contains, generalizes, improves, simplifies, and unifies many other results in this area in a natural way.

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Approximation schemes for scheduling on parallel machines

This investigation considers the problem of nonpreemptively assigning a set of independent tasks to a system of identical processors to maximize the earliest processor finishing time. While this goal is a nonstandard scheduling criterion, it does have natural applications in certain maintenance scheduling and deterministic fleet sizing problems. The problem is NP-hard, justifying an analysis of heuristics such as the well-known LPT algorithm in an effort to guarantee near-optimal results. It is proved that the worst-case performance of the LPT algorithm has an asymptotically tight bound of $\frac{4}{3}$ times the optimal.

Scheduling to maximize the minimum processor finish time in a multiprocessor system

We give a polynomial approximation scheme for the problem of scheduling on uniformly related parallel machines for a large class of objective functions that depend only on the machine completion times, including minimizing the lp norm of the vector of completion times. This generalizes and simplifies many previous results in this area.

Approximation Schemes for Scheduling on Uniformly Related and Identical Parallel Machines

The exact LPT-bound for maximizing the minimum completion time

This paper considers the problem of sequencing the installation of replacement parts for the repeated repair of a machine consisting of two working parts. A finite inventory of spares is initially available for each type of part. Each spare has a known deterministic field life, and spares of each type may have different field lives. The inventories are eventually depleted since no replacement or repairs of parts occur. The primary objective is to obtain a sequencing policy which minimizes the total number of installations replacements. This problem is a special case of many opportunistic replacement problems and is called a “build” problem. The motivation for this work is based on the maintenance of modular gas turbine aircraft engines used by domestic airlines and the U.S. Air Force. We show this build problem is NP-hard which indicates the necessity of efficient heuristic solution procedures. This paper considers several “greedy” type algorithms based on grouping parts according to exact matching of field lives. Then, a generalization is presented which allows approximate matches. The focus of the analysis is to obtain tight worst case bounds on the number of replacements obtained by each of the greedy algorithms.

D. K. Friesen

Papers

Analysis of Greedy Solutions for a Replacement Part Sequencing Problem